Markov evolutions and hierarchical equations in the continuum I. One-component systems

TL;DR

This paper develops a mathematical framework for describing the evolution of infinite particle systems in the continuum using Markov processes, deriving equations for correlation functions and illustrating with practical examples.

Contribution

It introduces a unified approach to derive evolution equations for correlation functions in continuum particle systems, applicable to various Markov dynamics.

Findings

Derived evolution equations for correlation functions and generating functionals.

Illustrated the framework with concrete examples of Markov evolutions.

Provided a general methodology applicable to infinite particle systems.

Abstract

General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications.